Abstract

We give a simple proof of the λ = d - 2 cases of the sharp Hardy-Littlewood-Sobolev inequality for d≥3, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for d = 2 via a monotone flow governed by the fast diffusion equation.

We explain an interesting relation between the sharp Hardy-Littlewood-Sobolev (HLS) inequality for the resolvent of the Laplacian, the sharp Gagliardo-Nirenberg-Sobolev (GNS) inequality, and the fast diffusion equation (FDE). As a consequence of this relation, we obtain an identity expressing the HLS functional as an integral involving the fast diffusion flow and the GNS functional. From this identity we obtain a simple proof of the sharp HLS inequality in the cases that express the regularizing properties of the Green’s function for the Laplacian in , for d≥3, and of the Logarithmic Hardy-Littlewood-Sobolev (Log-HLS) inequality, for d = 2. The proof also provides interesting information about the HLS functional that does not follow from previous proofs of the HLS inequality.

Throughout the paper, we shall use ‖f‖p to denote the usual Lp norms with respect to Lebesgue measure: , for 1 ≤ p < ∞.

The Sharp Hardy-Littlewood-Sobolev Inequality

The sharp form of the HLS inequality, due to Lieb (1), states that for all nonnegative measurable functions f on , and all 0 < λ < d, [1]where [2]and p = 2d/(2d - λ), and there is equality in [1] if and only if for some and , f is a nonzero multiple of h(x/s - x0).

The λ = d - 2 cases of the sharp HLS inequality are particularly interesting because they express the Lp smoothing properties of (-Δ)-1 on : for d≥3,[3]where |Sd-1| denotes the surface area of the d - 1 dimensional unit sphere in . The integrals on the right hand side of [1] can be computed explicitly in terms of Γ-functions, and, after some computation with the constants, one sees that for λ = d - 2, [1] can be rewritten as for all where [4]with [5]We refer to this functional on as the HLS functional.

Let g be any smooth function of compact support, and let 〈g,f〉 denote . Then the positivity of on implies that for all , Taking the supremum over f on both sides; i.e., computing two Legendre transforms, one finds [6]Notice that CS is the least constant for which [6] can hold for all smooth compactly supported functions g, because the Legendre transform can be undone so that any improvement in the constant in [6] would yield an improvement in the constant in the HLS inequality, and this is impossible.

We can summarize the last paragraph by saying the that sharp HLS inequality for λ = d - 2, d≥3, is dual to the sharp Sobolev inequality [6], and hence, once one knows the sharp constant to one of these inequalities, one knows the sharp constant to the other. A little thought shows that the same is true for optimizers, as well as sharp constants.

In this paper, we shall explain another kind of “duality” involving the λ = d - 2 cases of the HLS inequality. This duality relation, which does not have any evident connection with the Legendre transform argument explained above, relates the λ = d - 2 cases of the sharp HLS inequality to certain sharp Gagliardo-Nirenberg-Sobolev (GNS) inequalities, again with an identification of sharp constants and optimizers. The GNS inequalities in question are due, in their sharp form, to Del Pino and Dolbeault (2). These authors state that for all locally integrable functions f on , d≥2, with a square integrable distributional gradient, and p with 1 < p < d/(d - 2) [7]where [8]and θ = d(p - 1)/(p(d + 2 - (d - 2)p)). Moreover, there is equality in [7] if and only if for some and , f is a nonzero multiple of .

Notice that GNS optimizers are certain powers of HLS optimizers, and vice versa. In fact, there is yet another context in which the HLS optimizers play an important role: they are the steady-state solutions of certain nonlinear evolution equations pertaining to fast diffusion.

The Fast Diffusion Equation

The equation [9]with 0 < m < 1 describes fast diffusion. (For m = 1, it is the usual heat equation describing ordinary diffusion, and m > 1 corresponds to slow diffusion.) Note that u(x,t) solves Eq. 9 if and only if [10]with β = 2 - d(1 - m) satisfies the equation [11]For m = 1, this is the Fokker-Planck equation, and Eq. 11 is a nonlinear relative of the Fokker-Planck equation. For all 1 - 2/d < m < 1, the Eq. 11 has integrable stationary solutions. Computing them, one finds [12]The parameter D(M) fixes the massM of the steady-state v∞,M(x); i.e., the quantity Computing the integral one finds that where C(d,m,β) is a constant that may be expressed in terms of Γ-functions.

The self-similar solutions of Eq. 9 corresponding through the change of variables Eq. 10 to the v∞,M are known as Barenblatt solutions, and v∞,M is known as the Barenblatt profile for Eq. 9 with mass M. Notice in the limiting case m = 1, the Barenblatt profile approaches a Gaussian, as one would expect. The Barenblatt self-similar solutions are natural generalizations of the fundamental solutions of the heat equation. The Cauchy problem for the FDE Eq. 9 has been studied by many authors, we refer to ref. 3 for a full account of the literature.

It is established in ref. 4 that the range of mass conservation for the fast diffusion equation is 1 - 2/d < m < 1. As noted above, this range is exactly the range of m < 1 in which integrable self-similar solutions exist. Within this range, the flow associated to the fast diffusion equation is in many ways even better than the flow associated to the heat equation; see ref. 5 and the references therein. The solutions of Eq. 9 with positive integrable initial data are C∞ and strictly positive everywhere instantaneously, just as for the heat flow.

Moreover, for nonnegative initial data f of mass M satisfying [13]for some R > 0, which means that f is decaying at infinity at least as fast as the Barenblatt profile v∞,M, the solution v(x,t) of Eq. 9 with initial data f satisfies the following remarkable bounds: For any t∗ > 0, there exists a constant C = C(t∗) > 0 such that [14]for all t≥t∗ and . The lower bound is remarkable, as our assumption on the initial data is an upper bound. This lower bound shows “how fast” fast diffusion really is: it spreads mass out to infinity to produce the “right tails” instantly.

The proof of these bounds is based on the L∞-error estimate obtained in ref. 6 and improved to global Harnack inequalities in ref. 5, see also ref. 7. Moreover, one can show global smoothness estimates on the quotient, that is, for any t∗ > 0 [15]for all . Finally, it is well known that [16]For the best known rates of convergence see ref. 8.

Monotonicity of Along Fast Diffusion

Because the HLS minimizers are the attracting steady states for a certain fast diffusion flow, one might hope that the HLS functional would be monotone decreasing along this flow. This monotonicity is indeed the case.

Theorem 1.

Let be nonnegative, and suppose that f satisfies [13] for some R > 0, and m = d/(d + 2), ensuring in particular that f is integrable. Let us further suppose that [17]where h is given by Eq. 2 with λ = d - 2. Let u(x,t) be the solution of Eq. 9 with m = d/(d + 2) and u(x,1) = f(x). Then, for all t > 1, [18]where

[19]

Proof:

There are two things to be proved, namely the identity on the left hand side of Eq. 18, and also the nonnegativity of the functional defined in Eq. 19.

We begin with the identity. The uniform bounds on the regularity of the quotient [15] justify all of the integration-by-parts used in the following computation of the derivative of along the FDE flow for m = d/(d + 2): [20]Therefore, let u(x,t) solve Eq. 20. We compute [21]Now define g = u(d-1)/(d+2). Then one computes

Rewriting the right hand side of Eq. 21 in terms of g, one finds Expressing this identity in terms of , we have proved the left hand side of Eq. 18.

We shall now show that is nonnegative as a consequence of the p = (d + 1)/(d - 1) cases of the GNS inequality [7]. These cases can be written in the form [22]where, by definition, CGNS is the best constant for which this inequality is valid for all smooth g with compact support. One could compute the right hand side of [7] to determine the explicit value of CGNS and find that [23]An easier way to determine CGNS is to go back to the first part of the proof, and consider the initial data f = h, so that u(x,t) does not depend on t. Then by what we just proved, . Notice that h(d-1)/(d+2) is an optimizer for the case of [7]. Hence, for the optimal g, and this proves Eq. 23, and now the nonnegativity of follows from [22] and Eq. 23.

As we show in the next section, all of the information that we have used about the sharp GNS inequality can also be proved by a fast diffusion flow argument without bringing anything else into the argument. Thus, while at present, our analysis may not look self-contained, this will be remedied shortly. For now though, let us finish with the HLS inequality.

As a direct consequence of the previous theorem, we deduce an identity for the HLS functional that manifestly displays its nonnegativity.

Theorem 2.

Let , d≥3 be nonnegative. Suppose also that f satisfies [24]for some R > 0. Then with u defined as in Theorem 1, [25]Moreover, if and only if f is a multiple of h(x/s - x0) for some s > 0 and , with h given by Eq. 2, λ = d - 2.

Proof:

The assumption [24] together with the fact that implies the integrability of f. Because for all α > 0, , it is harmless to assume Eq. 17, which we do. We may now apply the previous theorem. Let v(x,t) be the solution of Eq. 11 with v(x,0) = f(x). Let u(x,t) be the solution of Eq. 9 with u(x,1) = f(x). Because of the scaling relation Eq. 10, we have with β = 4/(d + 2). Then Theorem 1 implies that, for all t > 0, [26]

We now claim that [27]Because , the latter fact is an easy consequence of the the global bounds in [14] due to the assumptions [13] and Eq. 17, and a dominated convergence argument. The former is slightly more subtle: First, it is easy to show, using known facts about the Cauchy problem for the FDE (3), that under our hypothesis, . By an argument using Fatou’s Lemma, the potential integral term can only jump downwards in the limit. Thus, at least we have , and hence integrating Eq. 26 over [0,∞), we obtain

In particular, we have shown that under the hypotheses of the theorem. Then an obvious truncation and monotone convergence argument using the sequence of function fn = min{f,nh} shows that is well defined, finite and nonnegative for all nonnegative . This analysis proves the λ = d - 2 HLS inequality, and then by a standard argument using the positive definite nature of the potential integral, shows that the potential integral is continuous on . Thus, is continuous on , and Eq. 27 now follows. Now integrating Eq. 26 over [0,∞) and using Eq. 27 yields the identity Eq. 25.

We now conclude from Eq. 25 that if and only if for all t, which is equivalent to the existence of a constant C and continuous functions s(t) and x0(t), defined for t > 1 such that due to the characterization of the optimizers in the GNS inequality [7]. Thus u(x,eβt) is at each t > 0 a Barenblatt profile, thus by uniqueness of the Cauchy problem for the FDE Eq. 9, u(x,eβt) is a self-similar Barenblatt solution of the FDE Eq. 9. Because in , hence f is itself a Barenblatt profile, meaning that f is a multiple of h(x/s - x0) for some s > 0 and some .

The identity [25] has been derived under the hypothesis [24]. However, it is easy to pass from Theorem 2 to to the following, which is simply a restatement of the λ = d - 2 cases of Lieb’s Theorem:

Theorem 3.

Let , d≥3 be nonnegative. Then , and if and only if f is a multiple of h(x/s - x0) for some s > 0 and some , and where h is given by Eq. 2, λ = d - 2.

Proof:

We have already proved the inequality in the course of proving Theorem 2. The cases of equality are somewhat more subtle, and are dealt with in the next lemma.

Lemma 1.

If is nonnegative and satisfies , then f satisfies [13] for some R > 0.

To prove Lemma 1 we make use, for the first time, of rearrangement inequalities and the conformal invariance of the HLS functional. Recently, Frank and Lieb gave an interesting proof of certain cases of the HLS inequality (9) that uses reflection positivity in place of rearrangements.

Proof of Lemma 1:

By a well known theorem of Lieb (10) on the cases of equality in the Riesz rearrangement inequality, every optimizer f in must be a translate of its spherically symmetric decreasing rearrangement, f∗. Making any necessary translation, we may assume f = f∗. Next, as also shown by Lieb, the HLS functional is invariant under the inversion mapping where , which is anisometry on . Letting x0 be any unit vector, f is uniformly bounded on the unit ball centered at 2x0. Thus |x|-(d+2)f(x/|x|2 - 2x0) is also an optimizer, and satisfies [13] for R = 1. Now the previous Theorem applies, and this function must be a Barenblatt profile. It follows that the same is true of the original f.

Note that we have only used the invariance of under inversion, and hence under the full conformal group, to settle the final points regarding cases of equality. It is remarkable that neither the fast diffusion flow, nor the GNS inequalities possess this invariance, and yet for a dense class of functions f, Eq. 25 expresses in terms of the fast diffusion flow and the GNS functional.

This proof by monotone flow cannot be applied to directly prove the Sobolev inequality [6] because the positive optimizers for it are not integrable, and thus are steady states of an equation of the form Eq. 11 with m < 1 - 2/d. In this range of m, the diffusion is so fast that mass is lost, see ref. 4; solutions of Eq. 11 with smooth compactly supported initial data become extinct in a finite time. In particular, neither [14] nor Eq. 16 are valid for such m, among other problems.

The Sharp GNS Inequalities and Fast Diffusion

As we have seen, a calculation using fast diffusion reduces the proof of the λ = d - 2 cases of the HLS inequality to the proof of certain GNS inequalities. We now show, using results in ref. 11, that another sort of calculation using a different fast diffusion reduces the proof these GNS inequalities to the Schwarz inequality.

The FDE Eq. 11 with 1 - 2/d < m < 1 is a gradient flow of the functional [28]with respect to the Euclidean Wasserstein distance, see ref. 14. In particular, is a Liapunov functional for Eq. 11, being its dissipation given by [29]for any solution v(·,t) to Eq. 11 and initial data v(x,0) satisfying the hypotheses of Theorem 2. Here, the regularity properties of the solution [14] and [15] that justified the computations in the previous section ensure that at least when the initial data satisfies [13], the dissipation of the entropy along the evolution is given by [30]with as shown in ref. 11. Define [31]By the Schwarz inequality for the Hilbert-Schmidt inner product, and thus from Eq. 30 and Eq. 31 we get [32]As long as (d - 1)/d < m < 1, the factor in front of is strictly positive.

Now combine Eq. 29 and [32] to conclude [33]Integrating this inequality in t from 0 to ∞, and using the fact that one gets [34]for all v(·,0) satisfying the assumptions of Theorem 2. Because , [35]This inequality is equivalent to the sharp GNS inequalities [7]. One sees inequality [35] by expanding the squares in this inequality, cancelling the second moment terms from both sides, and performing an integration-by-parts allowed by [14]. Then with m = (p + 1)/(2p) and the change of dependent variable v(x,0)≕f(x)2p, and finally using a standard scaling argument, one arrives at [7]; see ref. 2 for details. This sketch finishes the summary of the relevant results in refs. 2 and 11.

The exponent m of the FDE Eq. 9 used to prove the particular cases of the GNS inequalities involved in the proof of the HLS inequality in previous sections is m = d/(d + 1).

On the other hand, the exponent m in the FDE along which the HLS functional is monotone is m = d/(d + 2), which corresponds to the critical exponent of FDE related to the boundedness of the second moment of the stationary states v∞,M, and it plays a certain role in the large-time assymptotics of the FDE, see refs. 7 and 8.

We finally show how to extract from [33] the characterization of the optimizers of the GNS inequalities [7], at least under the conditions that are relevant for the application in the proof of Theorem 2.

Theorem 4.

Let f be a positive measurable function on , d≥2, with a square integrable distributional gradient f, such that [36]and f being an optimizer of the GNS inequality [7]. Then, f is given by Eq. 8 up to translations and dilations.

Proof:

Let us consider v(x,0) = f2p(x) as initial data for the FDE Eq. 11 with m = (p + 1)/2p. Under these conditions, we have derived [34], and because [35] is equivalent to the sharp GNS inequality for f and f is indeed a stationary state of the FDE Eq. 11 due to Eq. 12, it must be the case that Due to the positivity of v(·,t) for all t > 0, we conclude that Δξ = 0 for all t > 0. It is straightforward to infer from the global bounds [14] that for any t∗ > 0, there exists D1 = D1(t∗) > 0 such that for all t≥t∗, . Thus, ξ is a globally bounded harmonic function, and then Liouville’s theorem implies that ξ is constant. It follows that for each t, v(·,t) is a Barenblatt profile, which determines the form of f as in Theorem 2.

Proof of the Sharp Log-HLS Inequality via Fast Diffusion

In this section, we prove the sharp Log-HLS inequality on by a similar fast diffusion flow argument. The sharp Log-HLS inequality (12, 13) states that for all nonnegative measurable functions f on such that f ln f and f ln(e + |x|2) belong to , [37]where with C(M)≔M(1 + log π - log(M)). Moreover, there is equality if and only if f(x) = hγ,M(x - x0) for some γ > 0 and some , where [38]Note that all integrals figuring in the logarithmic HLS inequality are at least well defined with no cancellation of infinities in their sum under the condition that f ln f and f ln(e + |x|2) belong to .

We therefore define the Log-HLS functional by on the domain introduced above. The logarithmic HLS functional involves three distinct integral functionals of f while for d≥3, the HLS functional involves only two. A more significant difference is that the logarithmic HLS functional is invariant under scale changes. That is, for a > 0 and f in the domain of , define f(a)≔a2f(ax). One then computes that for all a > 0.

This scale invariance simplifies our application of the fast diffusion equation, to which we now turn. For d = 2, m = d/(d + 2) reduces to m = 1/2. Thus, the relevant cases of the fast diffusion equation in d = 2 are [39]and [40]As before, it is easily checked that the stationary states are given by for any mass M > 0 with suitably chosen D = D(M). Note that [41]for a suitable M∗. For d = 2, the scaling relation between these two equations is that u(x,t) solves Eq. 39 if and only if v(x,t)≔e2tu(etx,et) solves Eq. 40. Notice that with u and v related in this way, the scale invariance of implies that [42]

We now differentiate along the fast diffusion flow as before. For convenience, without loss of generality, we fix the initial mass. We also impose the appropriate version of [13].

Theorem 5.

Let f be a nonnegative measurable function on such that f ln f and f ln(e + |x|2) belong to . Suppose that , with h given by Eq. 41. Then , and there is equality if and only if f(x/s - x0) = h(x) for some s > 0 and some .

Suppose in addition that f satisfies [13] for some R > 0 and m = 1/2. Let u(x,t) be the solution of Eq. 39 with u(x,1) = f(x). Then we have the identity [43]where is nonnegative by the d = 2, p = 3 case of the sharp GNS inequality [7].

Proof:

Let v(x,t) be the solution of Eq. 40 with v(x,0) = f(x). Suppose initially that f ≤ Ch for some finite C. Under this additional argument is it easy to prove Eq. 27, though the t = 0 limit is more subtle because in d = 2, the potential integral is neither point wise positive, nor positive definite. However, our hypotheses ensure integrability of the positive and negative parts in the potential integral, and then monotone convergence may be used as before. A truncation argument, left to the reader, then removes the extra assumption f ≤ Ch.

Differentiability of is justified as before, and we have But by Eq. 42, . By the uniform regularity bounds on the quotient [15], we compute [44]Making the change of variables g = u1/4, , , and leading together with Eq. 44 to Eq. 43. The proof of the statement about cases of equality proceed exactly as in Theorem 2. The proof that the condition [13] may be relaxed as far as the inequality itself is concerned, is similar, with the integrability condition on f ln(e + |x|2) being used to insure integrability of the positive part of the potential integral.

Consequences of the Monotonicity

The monotonicity of the HLS and Log-HLS functionals along fast diffusion flows has interesting consequences. One consequence is the simplicity of the “landscape” of the Log-HLS functional: let be the (convex) set of nonnegative functions on satisfying all of the hypotheses of Theorem 5. Let be Log-HLS functional on . Then there are no strict local minimizers of in other than the absolute minimizers.

Indeed, this conclusion is an immediate consequence of Theorems 1 and 5: One can go monotonically down to the absolute minimizers from any point in . Of course, a similar result holds for the HLS funcional, but in this case the Euler-Lagrange equation is thoroughly studied, and there are no nonnegative critical points apart form global minimizers.

Next, as we have noted, the fast diffusion flow along which we have shown , corresponding to the Log-HLS inequality in d = 2, to be monotone decreasing is gradient flow in the 2-Wasserstein metric for the entropy functional defined in Eq. 28. There is a kind of duality between the HLS functional and the entropy functional , as we now explain.

We first recall an observation of Matthes, McCann, and Savare (15) concerning pairs of gradient flow equations. Consider two smooth functions Φ and Ψ on , and consider the two ordinary differential equations describing gradient flow: Then of course Φ[(x(t)] and Ψ[(y(t)] are monotone decreasing. Now differentiate each function along the other’s flow: [45]Thus, Φ is decreasing along the gradient flow of Ψ for any initial data if and only if Ψ is decreasing along the gradient flow of Φ for any initial data.

An analog of this argument holds for well behaved gradient flows in the 2-Wasserstein sense, which is the result of ref. 15. In our case, we can apply it to the Log-HLS functional in d = 2. Thus, because for the Log-HLS functional is decreasing along the 2-Wasserstein gradient flow for , one can expect that is decreasing along the 2-Wasserstein gradient flow for , which turns out to be nothing other than the critical mass case of the Patlak-Keller-Segel equation. Actually, the m = 1/2, d = 2 version of must be “renormalized” because in this case v∞,M does not have finite second moments, nor an integrable square root. Still, this “second Lyapunov function” has been shown to be very useful in analyzing the critical mass Patlak-Keller-Segel equation; see ref. 16.

Finally, the main results here, namely, the integral identities of Eqs. 25 and 43, provide the starting point of an analysis of “remainder terms” and “stability” results for the sharp HLS and Log-HLS inequalities. A quantitative stability theorem shall be developed elsewhere.

Acknowledgments

E.A.C. was partially supported by National Science Foundation Grant DMS 0901632. J.A.C. acknowledges support from the project MTM2008-06349-C03-03 DGI-MCI (Spain) and 2009-SGR-345 from Agència de Gestió d'Ajuts Universitaris i de Recerca (AGAUR)-Generalitat de Catalunya. M.L. was partially supported by National Science Foundation Grant DMS 0901304.

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